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Theorem uvtx01vtx 26020
 Description: If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. This theorem could have been stated ((𝑉 UnivVertex ∅) ≠ ∅ ↔ (#‘𝑉) = 1), but a lot of auxiliary theorems would have been needed. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
uvtx01vtx ((𝑉 UnivVertex ∅) ≠ ∅ ↔ ∃𝑥 𝑉 = {𝑥})
Distinct variable group:   𝑥,𝑉

Proof of Theorem uvtx01vtx
Dummy variables 𝑒 𝑘 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4718 . . . . 5 ∅ ∈ V
2 isuvtx 26016 . . . . 5 ((𝑉 ∈ V ∧ ∅ ∈ V) → (𝑉 UnivVertex ∅) = {𝑥𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅})
31, 2mpan2 703 . . . 4 (𝑉 ∈ V → (𝑉 UnivVertex ∅) = {𝑥𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅})
43neeq1d 2841 . . 3 (𝑉 ∈ V → ((𝑉 UnivVertex ∅) ≠ ∅ ↔ {𝑥𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅} ≠ ∅))
5 rabn0 3912 . . . 4 ({𝑥𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅} ≠ ∅ ↔ ∃𝑥𝑉𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅)
65a1i 11 . . 3 (𝑉 ∈ V → ({𝑥𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅} ≠ ∅ ↔ ∃𝑥𝑉𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅))
7 df-rex 2902 . . . 4 (∃𝑥𝑉𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅ ↔ ∃𝑥(𝑥𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅))
8 noel 3878 . . . . . . . . . 10 ¬ {𝑘, 𝑥} ∈ ∅
9 rn0 5298 . . . . . . . . . . 11 ran ∅ = ∅
109eleq2i 2680 . . . . . . . . . 10 ({𝑘, 𝑥} ∈ ran ∅ ↔ {𝑘, 𝑥} ∈ ∅)
118, 10mtbir 312 . . . . . . . . 9 ¬ {𝑘, 𝑥} ∈ ran ∅
1211ralf0 4030 . . . . . . . 8 (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅ ↔ (𝑉 ∖ {𝑥}) = ∅)
1312a1i 11 . . . . . . 7 (𝑉 ∈ V → (∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅ ↔ (𝑉 ∖ {𝑥}) = ∅))
1413anbi2d 736 . . . . . 6 (𝑉 ∈ V → ((𝑥𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅) ↔ (𝑥𝑉 ∧ (𝑉 ∖ {𝑥}) = ∅)))
15 ssdif0 3896 . . . . . . . . 9 (𝑉 ⊆ {𝑥} ↔ (𝑉 ∖ {𝑥}) = ∅)
1615a1i 11 . . . . . . . 8 (𝑉 ∈ V → (𝑉 ⊆ {𝑥} ↔ (𝑉 ∖ {𝑥}) = ∅))
1716bicomd 212 . . . . . . 7 (𝑉 ∈ V → ((𝑉 ∖ {𝑥}) = ∅ ↔ 𝑉 ⊆ {𝑥}))
1817anbi2d 736 . . . . . 6 (𝑉 ∈ V → ((𝑥𝑉 ∧ (𝑉 ∖ {𝑥}) = ∅) ↔ (𝑥𝑉𝑉 ⊆ {𝑥})))
19 sssn 4298 . . . . . . . . 9 (𝑉 ⊆ {𝑥} ↔ (𝑉 = ∅ ∨ 𝑉 = {𝑥}))
2019anbi2i 726 . . . . . . . 8 ((𝑥𝑉𝑉 ⊆ {𝑥}) ↔ (𝑥𝑉 ∧ (𝑉 = ∅ ∨ 𝑉 = {𝑥})))
21 n0i 3879 . . . . . . . . . . . 12 (𝑥𝑉 → ¬ 𝑉 = ∅)
2221pm2.21d 117 . . . . . . . . . . 11 (𝑥𝑉 → (𝑉 = ∅ → 𝑉 = {𝑥}))
2322imp 444 . . . . . . . . . 10 ((𝑥𝑉𝑉 = ∅) → 𝑉 = {𝑥})
24 simpr 476 . . . . . . . . . 10 ((𝑥𝑉𝑉 = {𝑥}) → 𝑉 = {𝑥})
2523, 24jaodan 822 . . . . . . . . 9 ((𝑥𝑉 ∧ (𝑉 = ∅ ∨ 𝑉 = {𝑥})) → 𝑉 = {𝑥})
26 vsnid 4156 . . . . . . . . . . 11 𝑥 ∈ {𝑥}
27 eleq2 2677 . . . . . . . . . . 11 (𝑉 = {𝑥} → (𝑥𝑉𝑥 ∈ {𝑥}))
2826, 27mpbiri 247 . . . . . . . . . 10 (𝑉 = {𝑥} → 𝑥𝑉)
29 olc 398 . . . . . . . . . 10 (𝑉 = {𝑥} → (𝑉 = ∅ ∨ 𝑉 = {𝑥}))
3028, 29jca 553 . . . . . . . . 9 (𝑉 = {𝑥} → (𝑥𝑉 ∧ (𝑉 = ∅ ∨ 𝑉 = {𝑥})))
3125, 30impbii 198 . . . . . . . 8 ((𝑥𝑉 ∧ (𝑉 = ∅ ∨ 𝑉 = {𝑥})) ↔ 𝑉 = {𝑥})
3220, 31bitri 263 . . . . . . 7 ((𝑥𝑉𝑉 ⊆ {𝑥}) ↔ 𝑉 = {𝑥})
3332a1i 11 . . . . . 6 (𝑉 ∈ V → ((𝑥𝑉𝑉 ⊆ {𝑥}) ↔ 𝑉 = {𝑥}))
3414, 18, 333bitrd 293 . . . . 5 (𝑉 ∈ V → ((𝑥𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅) ↔ 𝑉 = {𝑥}))
3534exbidv 1837 . . . 4 (𝑉 ∈ V → (∃𝑥(𝑥𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅) ↔ ∃𝑥 𝑉 = {𝑥}))
367, 35syl5bb 271 . . 3 (𝑉 ∈ V → (∃𝑥𝑉𝑘 ∈ (𝑉 ∖ {𝑥}){𝑘, 𝑥} ∈ ran ∅ ↔ ∃𝑥 𝑉 = {𝑥}))
374, 6, 363bitrd 293 . 2 (𝑉 ∈ V → ((𝑉 UnivVertex ∅) ≠ ∅ ↔ ∃𝑥 𝑉 = {𝑥}))
38 id 22 . . . . . . 7 𝑉 ∈ V → ¬ 𝑉 ∈ V)
3938intnanrd 954 . . . . . 6 𝑉 ∈ V → ¬ (𝑉 ∈ V ∧ ∅ ∈ V))
40 df-uvtx 25951 . . . . . . 7 UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒})
4140mpt2ndm0 6773 . . . . . 6 (¬ (𝑉 ∈ V ∧ ∅ ∈ V) → (𝑉 UnivVertex ∅) = ∅)
4239, 41syl 17 . . . . 5 𝑉 ∈ V → (𝑉 UnivVertex ∅) = ∅)
4342notnotd 137 . . . 4 𝑉 ∈ V → ¬ ¬ (𝑉 UnivVertex ∅) = ∅)
44 df-ne 2782 . . . 4 ((𝑉 UnivVertex ∅) ≠ ∅ ↔ ¬ (𝑉 UnivVertex ∅) = ∅)
4543, 44sylnibr 318 . . 3 𝑉 ∈ V → ¬ (𝑉 UnivVertex ∅) ≠ ∅)
46 snex 4835 . . . . . 6 {𝑥} ∈ V
47 eleq1 2676 . . . . . 6 (𝑉 = {𝑥} → (𝑉 ∈ V ↔ {𝑥} ∈ V))
4846, 47mpbiri 247 . . . . 5 (𝑉 = {𝑥} → 𝑉 ∈ V)
4948exlimiv 1845 . . . 4 (∃𝑥 𝑉 = {𝑥} → 𝑉 ∈ V)
5049con3i 149 . . 3 𝑉 ∈ V → ¬ ∃𝑥 𝑉 = {𝑥})
5145, 502falsed 365 . 2 𝑉 ∈ V → ((𝑉 UnivVertex ∅) ≠ ∅ ↔ ∃𝑥 𝑉 = {𝑥}))
5237, 51pm2.61i 175 1 ((𝑉 UnivVertex ∅) ≠ ∅ ↔ ∃𝑥 𝑉 = {𝑥})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  ran crn 5039  (class class class)co 6549   UnivVertex cuvtx 25948 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-uvtx 25951 This theorem is referenced by: (None)
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